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Determinantal process starting from an orthogonal symmetry is a Pfaffian process. (English) Zbl 1235.82016

Summary: When the number of particles \(N\) is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index \(\nu > -1\) (\(\text{BESQ}^{(\nu)}\)) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The \(2 \times 2\) skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin \(N\delta _{0}\) and by the equivalence between the noncolliding \(\text{BESQ}^{(\nu)}\) and that of the noncolliding squared generalized meander starting from \(N\delta_{0}\).

MSC:

82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
60J65 Brownian motion
15B52 Random matrices (algebraic aspects)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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